All of the 5th grade teachers and students from Springer went on a field trip to an art museum. Tickets were $$8.50$ each for teachers and $$3.00$ each for students, and the group paid $$61.00$ in total. The next month, the same group visited a natural history museum where the tickets cost $$34.00$ each for teachers and $$12.50$ each for students, and the group paid $$248.50$ in total. Find the number of teachers and students on the field trips.
Answer: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${8.5x+3y = 61}$ ${34x+12.5y = 248.5}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-4$ ${-34x-12y = -244}$ ${34x+12.5y = 248.5}$ Add the top and bottom equations together. $ 0.5y = 4.5 $ $ y = \dfrac{4.5}{0.5}$ ${y = 9}$ Now that you know ${y = 9}$ , plug it back into $ {8.5x+3y = 61}$ to find $x$ ${8.5x + 3}{(9)}{= 61}$ $8.5x+27 = 61$ $8.5x = 34$ $x = \dfrac{34}{8.5}$ ${x = 4}$ You can also plug ${y = 9}$ into $ {34x+12.5y = 248.5}$ and get the same answer for $x$ ${34x + 12.5}{(9)}{= 248.5}$ ${x = 4}$ There were $4$ teachers and $9$ students on the field trips.